The distances to the nearest stars can be measured by trigonometric parallax. A star with a parallax of one arc second (1”) is 1 parsec—about 3.3 light-years—away. Stars have real motion through space as well as apparent motion as Earth orbits the Sun. A star’s proper motion, its true motion across the sky, is a measure of the star’s velocity perpendicular to our line of sight. The star’s radial velocity—along the line of sight—is measured by the Doppler shift of its spectral lines.
The apparent brightness of a star is the rate at which energy from the star reaches unit area of a detector. Apparent brightness falls off as the inverse square of the distance. Optical astronomers use the magnitude scale to express and compare stellar brightnesses. The greater the magnitude, the fainter the star; a difference of five magnitudes corresponds to a factor of 100 in brightness. Apparent magnitude is a measure of apparent brightness. The absolute magnitude of a star is the apparent magnitude it would have if placed at a standard distance of 10 pc from the viewer. It is a measure of the star’s luminosity.
Astronomers often measure the temperatures of stars by measuring their brightnesses through two or more optical filters, then fitting a blackbody curve to the results. The color index of a star is the difference in its apparent magnitudes measured through two standard filters. The measurement of the amount of starlight received through each of a set of filters is called photometry. Astronomers classify stars according to the absorption lines in their spectra. The lines seen in the spectrum of a given star depend mainly on its temperature, and spectroscopic observations of stars provide an accurate means of determining both stellar temperatures and stellar composition. The standard stellar spectral classes, in order of decreasing temperature, are O, B, A, F, G, K, and M.
Only a few stars are large enough and close enough that their radii can be measured directly. The sizes of most stars are estimated indirectly through the radius–luminosity–temperature relationship. Stars are categorized as dwarfs comparable in size to or smaller than the Sun, giants up to 100 times larger than the Sun, and supergiants more than 100 times larger than the Sun. In addition to “normal” stars such as the Sun, two other important classes of star are red giants, which are large, cool, and luminous, and white dwarfs, which are small, hot, and faint.
A plot of stellar luminosities versus stellar spectral classes (or temperatures) is called an H–R diagram, or a color–magnitude diagram. About 90 percent of all stars plotted on an H–R diagram lie on the main sequence, which stretches from hot, bright blue supergiants and blue giants, through intermediate stars such as the Sun, to cool, faint red dwarfs. Most main-sequence stars are red dwarfs; blue giants are quite rare. About 9 percent of stars are in the white dwarf region, and the remaining 1 percent are in the red giant region.
By careful spectroscopic observations, astronomers can determine a star’s luminosity class, allowing them to distinguish main-sequence stars from red giants or white dwarfs of the same spectral type (or color). Once a star is known to be on the main sequence, measurement of its spectral type allows its luminosity to be estimated and its distance to be measured. This method of distance determination, which is valid for stars up to several thousand parsecs from Earth, is called spectroscopic parallax.
Most stars are not isolated in space but instead orbit other stars in binary-star systems. In a visual binary, both stars can be seen and their orbit charted. In a spectroscopic binary, the stars cannot be resolved, but their orbital motion can be detected spectroscopically. In an eclipsing binary, the orbit is oriented in such a way that one star periodically passes in front of the other as seen from Earth and dims the light we receive. The binary’s light curve is a plot of its apparent brightness as a function of time.
Studies of binary-star systems often allow stellar masses to be measured. The mass of a star determines its size, temperature, and brightness. Fairly well defined mass–radius and mass–luminosity relations exist for main-sequence stars. Hot blue giants are much more massive than the Sun; cool red dwarfs are much less massive. The lifetime of a star can be estimated by dividing its mass by its luminosity. High-mass stars burn their fuel rapidly and have much shorter lifetimes than the Sun. Low-mass stars consume their fuel slowly and may remain on the main sequence for trillions of years.
|PROBLEMS||Algorithmic versions of these questions are available in the Practice Problems Module of the Companion Website.|
The number of squares preceding each problem indicates the approximate level of difficulty.
1. How far away is the star Spica, whose parallax is 0.013''? What would Spica’s parallax be if it were measured from an observatory on Neptune’s moon Triton as Neptune orbited the Sun? HINT
2. A star lying 20 pc from the Sun has proper motion of 0.5''/yr. What is its transverse velocity? If the star’s spectral lines are observed to be redshifted by 0.01 percent, calculate the magnitude of its three-dimensional velocity relative to the Sun. HINT
3. What is the luminosity of a star having three times the radius of the Sun and a surface temperature of 10,000 K? HINT
4. A certain star has a temperature twice that of the Sun and a luminosity 64 times greater than the solar value. What is its radius, in solar units? HINT
5. Two stars—A and B, of luminosities 0.5 and 4.5 times the luminosity of the Sun, respectively—are observed to have the same apparent brightness. Which one is more distant, and how much farther away is it than the other? HINT
6. Two stars—A and B, of absolute magnitudes 3 and 8, respectively—are observed to have the same apparent magnitude. Which one is more distant, and how much farther away is it than the other? HINT
7. Calculate the solar energy flux (energy received per unit area per unit time), as seen from a distance of 10 pc from the Sun. Compare it with the solar constant at Earth. HINT
8. Astronomical objects visible to the naked eye range in apparent brightness from faint sixth-magnitude stars to the Sun with magnitude -27. What range in energy flux corresponds to this magnitude range? HINT
9. A star has apparent magnitude 4.0 and distance 100 pc. What is its absolute magnitude? HINT
10. A star has apparent magnitude 10.0 and absolute magnitude 2.5. How far away is it? HINT
11. Using the data shown in Figure 17.7, calculate the greatest distance at which a star like the Sun could be seen using (a) binoculars, (b) a typical 1-m telescope, (c) a 4-m telescope, and (d) the Hubble Space Telescope. HINT
12. In an eclipsing binary, the brightness is observed to drop by 1 percent when the brighter component eclipses the fainter one, and by 10 percent when the fainter component eclipses the brighter one (see Figure 17.21). If the stars lie on the main sequence and their radii follow the law R M, what is the ratio of the smaller mass to the larger one? HINT
13. Two stars in an eclipsing spectroscopic binary are observed to have an orbital period of 25 days. Further observations reveal that the orbit is circular, with a separation of 0.3 A.U., and that one star is 1.5 times the mass of the other. What are the masses of the stars? HINT
14. Given that the Sun’s lifetime is about 10 billion years, estimate the life expectancy of (a) a 0.2-solar mass, 0.01-solar luminosity red dwarf (b) A 3-solar mass, 30-solar luminosity star, (c) A 10-solar mass, 1000-solar luminosity blue giant. HINT
15. Assuming the mass–luminosity relation L M4, estimate the mass of the faintest main-sequence star that could be observed at a distance of 50,000 pc by (a) a typical 1-m telescope and (b) the Hubble Space Telescope. (See Figure 17.7.) HINT
1. Inverse-Square Law. Considering where your group is sitting right now, how many times dimmer would an imaginary, super-deluxe, ultra bright flashlight be if it were located at the front door of the group member who lives farthest away as compared to if it were at the front door of the group member who lives closest? Explain your reasoning.
2. Differences in Brightness. As a group, go to Appendix 3: Table 4—The Twenty Brightest Stars, and select two stars. Refer to the table column providing the apparent visual magnitude of the stars and compare how many times brighter one is as compared to the other.
3. Spectral Class and Temperature. Each group member should select a different star listed in Appendix 3: Table 4—The Twenty Brightest Stars, and estimate its temperature based on its spectral class. Explain your reasoning.
RESEARCHING ON THE WEB
To complete the following exercises, go to the online Destinations Module for Chapter 17 on the Companion Website for Astronomy Today 4/e.
1. Access the "All-Star Line Up" pages and rank the following stars from largest to smallest in radius: Betelgeuse, Mu Cephei, 40 Eridani, and Aldebaran.
2. Access the "Constellations and Their Stars" pages and determine the names of the three brightest stars in your horoscope birth sign.
3. Access the "Measuring the Brightness of Stars" page and describe the differences in star brightness that can be detected by the human eye.
1. Every winter, you can find an astronomy lesson in the evening sky. The Winter Circle is an asterism—or pattern of stars—made up of six bright stars in five different constellations: Sirius, Rigel, Betelgeuse, Aldebaran, Capella, and Procyon. These stars span nearly the entire range of colors (and therefore temperatures) possible for normal stars. Rigel is a B-type star, Sirius is an A-type, Procyon is an F-type star, Capella is a G-type star, Aldebaran a K-type star, and Betelgeuse is an M-type star. The color differences of these stars are easy to see. Why do you suppose there is no O-type star in the Winter Circle?
2. In the winter sky, you’ll find the red supergiant Betelgeuse in the constellation Orion. It’s easy to see because it’s one of the brightest stars visible in our night sky. Betelgeuse is a variable star with a period of about 6.5 years. Its brightness changes as it expands and contracts. At maximum size, Betelgeuse fills a volume of space that would extend from the Sun to beyond the orbit of Jupiter. Betelgeuse is thought to be about 10 to 15 times more massive than our Sun, and probably between four and 10 million years old. A similar star can be found shining prominently in midsummer. This is the red supergiant Antares in the constellation Scorpius. Depending on the time of year, can you find one of these stars? Why are they red?
|SKYCHART III PROJECTS||The SkyChart III Student Version planetarium program on which these exercises are based is included as a separately executable program on the CD in the back of this text.|
1. To see examples of the relationship between the B–V color index of a star and its color, locate the following bright stars using SkyChart III. The stars are arranged from hottest to coolest. Make a table with a column for the B–V color index and another column for the color the star appears to your eyes when it is displayed with SkyChart III. It will be easiest to locate the objects with SkyChart III by setting view for a 5° field, and using VIEW/Center Object to place each star in the center of the screen. Click on the star and the Object Info box will appear. In that box you will find details, such as color index, on the star. Deselect the horizon mask.
While the color of each star presented on your computer screen may be quite accurate, nothing beats seeing the actual star. Make a third column for the color you perceive it to be when you observe it in the sky, either through a telescope or with the unaided eye. On a clear night observe the stars yourself and relate their color to the color index. A telescope is not necessary for this observation, though a pair of binoculars might help, particularly if you are near city lights. You will find it necessary to work with the date and time settings of your simulation to determine the time of year each star is available to you at a time convenient for observation.
2. Simulate the view through a telescope of the binary pair Albireo and its companion using SkyChart III. Locate Albireo in Cygnus, or more conveniently with VIEW/Center Object/Albireo. Choose a field of approximately 1°. Left click on each of the stars of the binary pair to obtain the Object Info table. Caution: the Albireo dot is really two close stars (view at 1/120º field to see both). Reclick on another part of the dot if the Albireo statistics do not appear. Note the relationship between color index and the apparent color of each star. When the opportunity presents itself, view this famous binary pair with a telescope and observe the actual colors. When you see these two jewels through a telescope you will appreciate why they are so popular with amateur astronomers.
To simulate the image through a telescope, select TELESCOPE/Display. Deselect Camera field width. Set Finder field width to 3° and Eyepiece field width to 1°. Under TELESCOPE/Connect select Emulate Real Telescope Connection. Close this window and a Telescope Control box should open. Click on Albireo and click on Center in the Object Info box. Then go to the Telescope Control and click on Go To. The concentric set of rings will slew over to center on Albireo. The center ring presents the image seen through a typical amateur telescope. Zoom out to 5° and you will see the image typically seen through the finder scope depicted by the larger circle.
3. For this exercise you will determine the number of stars visible to the naked eye. You will take several samples and determine the average number of stars per sample. By imaging the stars as all contained within a celestial sphere, the following formula can be used to determine the total number of stars:
Your sample areas only represent a fraction of the total celestial sphere; “0.55cos(R)”, where R is the radius in degrees of your sample area, represents this fraction of the whole. Choose at least 10 sample areas, making a common star like Vega the center of your sample (this makes repeating the exercise much easier). Next you will use the telescope controls to define your sample area. Select TELESCOPE/Display and set the Finder field width to 4°, and deselect the Eyepiece field width and Camera field width. Under TELESCOPE/Connect select Emulate Real Telescope Connection and click OK. The Telescope Control box will open. Use the Find Object command in the EDIT menu to Select and Center on the star you are looking for. Then go to the Telescope Control box and click on Go To. After a short time a circle will appear centered around the star you have chosen with a radius of 2°. The last step before counting the stars in the sample area is to adjust the visible limits to roughly match that of a person viewing from a typical suburban area. Select DRAW/Symbols & grids. In the upper left corner of the window, change Faintest magnitude to 6.0 (check the status bar to make sure 6.0 remains the limit, because if you zoom in or out, the limit may change). About how many visible stars are there?
4. Repeat the preceding exercise using the exact same sample areas but adjust the visible limits to match that of an average pair of binoculars, a limit of about 10. About how many more stars can be seen with a pair of binoculars than with the naked eye?
5. Measure parallax. The parallax for all stars is less than 1 arc second or all stars are farther away than 1 pc (see Chapter 1). However, if you were viewing from Pluto, the parallax would increase by roughly a factor of 40. From the planet Pluto, follow Eridani (find “Epsilon Eri”) over 250 years by turning on trails and select COMPUTATION/Precision. In the window that appears, select Correct for Parallax and set the distance to 1.0. Explain your observations.
In addition to the Practice Problems and Destinations modules, the Companion Website at http://www.prenhall.com/chaisson
provides for each chapter an additional true-false, multiple choice,
and labeling quiz, as well as additional annotated images, animations,
and links to related Websites.